Oldham分形链与Liu-Kaplan分形链分抗的阻纳函数求解
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摘要
针对Oldham RC分形链类的电路特征,给定初始阻抗,采用3种方法理论推导Oldham分形链类阻抗函数解析表达式,并对比分析各求解方法。根据Oldham分形链分抗逼近电路的连分式表示和连分式三项递推公式,引入阻抗函数新的数学表示形式:连分式三项递推矩阵。通过分析Liu-Kaplan标度迭代电路和标度方程,推导出2种Liu-Kaplan分形链类阻抗函数的数学表示形式。通过理论验证和实验仿真对比不同分数阶下的阻抗函数表达式和频域特征与运算特征曲线。
A iming for the circuit characteristics of the Oldham RC fractal chain, the analytical expressions of Oldham RC fractal chain impedance function are deduced by using three methods at given initial impedances. Then, the solution methods are compared and analyzed. According to the continued fractional representation of the approximation circuit of Oldham fractal chai n fractance, and the three-term recurrence formula of continuous fraction, a new mathematical representation of impedance function is introduced: continuous fractional three-term recursion matrix. By analyzing the Liu-Kaplan scaling iteration circuit and t he scaling equations, two mathematical representations of the impedance functions of Liu-Kaplan fractal chains are deduced. Theoretical and experimental simulations compare the impedance function expressions and frequency characteristics of differen t fractional orders.
A iming for the circuit characteristics of the Oldham RC fractal chain, the analytical expressions of Oldham RC fractal chain impedance function are deduced by using three methods at given initial impedances. Then, the solution methods are compared and analyzed. According to the continued fractional representation of the approximation circuit of Oldham fractal chai n fractance, and the three-term recurrence formula of continuous fraction, a new mathematical representation of impedance function is introduced: continuous fractional three-term recursion matrix. By analyzing the Liu-Kaplan scaling iteration circuit and t he scaling equations, two mathematical representations of the impedance functions of Liu-Kaplan fractal chains are deduced. Theoretical and experimental simulations compare the impedance function expressions and frequency characteristics of differen t fractional orders.
引文
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[8]袁子,袁晓.规则RC分形分抗逼近电路的零极点分布[J].电子学报, 2017,45(10):2511-2520.(YUAN Zi,YUAN Xiao.Zero-pole distribution of regular RC fractal fractance app roximation circuit[J]. Chinese Journal of Electronics, 2017,45(10):2511-2520.)
[9]易舟,袁晓.分形分抗逼近电路零极点的数值求解与验证[J].太赫兹科学与电子信息学报, 2017,15(1):98-103.(YI Zhou,YUAN Xiao. The numerical solution and verificatio n of zero-pole for some fractal fractance appro ximation circuits[J].Journal of Terahertz Science and Electronic Information Technology, 2017,15(1):98-103.)
[10]LIU S H. Fractal model for the ac response of rough interface[J].Physical Review Letters, 1985, 55(5):529-532.
[11]KAPLAN T,GRAY L J. Effect of disorder on a fractal model f or the ac respons e of a rough interface[J]. Physical Review Letters, 1985,55(5):529-532.
[12]KAPLAN T,GRAY L J, L IU S H. Sel f-af f i ne f ract al model for a metal-electrolyte int erface[J]. Physi cal Revi ew Let t ers,1987,35(10):5379-5381.
[13]JONES W B,THRON W J,BROWDER F E,et al. Continued fractions:analytic theory and applications[M]. Massachusetts:Addison-Wesley Publishing Company, 1980.
[14]檀结庆.连分式理论及其应用[M].北京:科学出版社, 2007.(TAN Jieqing. Fractional theory and its applications[M].Beijing:Science Press, 2007.)
[15]数学手册编写组.数学手册[M].北京:人民教育出版社, 1979.(Editing group of mathematics manual. Math manual[M].Beijing:People's Education Press, 1979.)
[2]GOTO M,OLDHAM K B. Semi-integral electro-analysis:shapes of neopolarograms[J]. Analytical Chemistry, 1973, 45(12):2043-2050.
[3]OLDHAM K B. Semi-integral electro-analysis:analog implementatio n[J]. Analytical Chemistry, 1973,45(1):39-47.
[4]OLDHAM K B,SPANIER J. The fractional calculus:theory and applications of differentiation and integration to arbitrary order[M]. New York:Academic Press, 1974.
[5]袁晓.分抗逼近电路之数学原理[M].北京:科学出版社, 2015.(YUAN X i ao. Mat hemat i cal pri nci pl es of f ract ance approximation circuits[M]. Beijing:Science Press, 2015.)
[6]袁晓,冯国英. Oldham分形链分抗类与新型Liu-Kaplan标度方程[C]//中国电子学会电路与系统分会第二十六届学术年会.长沙,湖南:[s.n.], 2015:295-300.(YUAN Xiao,FENG Guoying. Oldham fractal chain class and new Liu-Kaplan scale equation[C]//The Symposium of the 26th Academ ic Annual Meeting of the Circuit and System Branch of the Chinese Institute of Electronics. Changsha,Hunan,China:[s.n.], 2015:295-300.)
[7]袁晓,冯国英.粗糙界面电极的电路建模与Liu-Kaplan标度方程[C]//中国电子学会电路与系统分会第二十六届学术年会.长沙,湖南:[s.n.], 2015:140-148.(YUAN Xiao,FENG Guoying. Circuit modeling of rough interface electrodes and Liu-Kaplan scalingequation[C]//The Symposium of the 26th Academic An nualMeetingof the Circuit andSystem Branch of the Chinese Institute of Electronics.Changsha,Hunan,China:[s.n.], 2015:140-148.)
[8]袁子,袁晓.规则RC分形分抗逼近电路的零极点分布[J].电子学报, 2017,45(10):2511-2520.(YUAN Zi,YUAN Xiao.Zero-pole distribution of regular RC fractal fractance app roximation circuit[J]. Chinese Journal of Electronics, 2017,45(10):2511-2520.)
[9]易舟,袁晓.分形分抗逼近电路零极点的数值求解与验证[J].太赫兹科学与电子信息学报, 2017,15(1):98-103.(YI Zhou,YUAN Xiao. The numerical solution and verificatio n of zero-pole for some fractal fractance appro ximation circuits[J].Journal of Terahertz Science and Electronic Information Technology, 2017,15(1):98-103.)
[10]LIU S H. Fractal model for the ac response of rough interface[J].Physical Review Letters, 1985, 55(5):529-532.
[11]KAPLAN T,GRAY L J. Effect of disorder on a fractal model f or the ac respons e of a rough interface[J]. Physical Review Letters, 1985,55(5):529-532.
[12]KAPLAN T,GRAY L J, L IU S H. Sel f-af f i ne f ract al model for a metal-electrolyte int erface[J]. Physi cal Revi ew Let t ers,1987,35(10):5379-5381.
[13]JONES W B,THRON W J,BROWDER F E,et al. Continued fractions:analytic theory and applications[M]. Massachusetts:Addison-Wesley Publishing Company, 1980.
[14]檀结庆.连分式理论及其应用[M].北京:科学出版社, 2007.(TAN Jieqing. Fractional theory and its applications[M].Beijing:Science Press, 2007.)
[15]数学手册编写组.数学手册[M].北京:人民教育出版社, 1979.(Editing group of mathematics manual. Math manual[M].Beijing:People's Education Press, 1979.)
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